Normal fluctuation in quantum ergodicity for Wigner matrices

Giorgio Cipolloni; László Erdős; Dominik Schröder

Normal fluctuation in quantum ergodicity for Wigner matrices

Probability 2022-03-04 v3 Mathematical Physics math.MP

Authors: Giorgio Cipolloni , László Erdős , Dominik Schröder

Abstract

We consider the quadratic form of a general deterministic matrix on the eigenvectors of an $N\times N$ Wigner matrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the large $N$ limit. The proof is a combination of the energy method for the Dyson Brownian motion inspired by [Marcinek, Yau 2020] and our recent multi-resolvent local laws [Cipolloni, Erd\H{o}s, Schr\"oder 2020].

Keywords

fluctuation theory random matrix theory gaussian random fields

Cite

@article{arxiv.2103.06730,
  title  = {Normal fluctuation in quantum ergodicity for Wigner matrices},
  author = {Giorgio Cipolloni and László Erdős and Dominik Schröder},
  journal= {arXiv preprint arXiv:2103.06730},
  year   = {2022}
}

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24 pages

Normal fluctuation in quantum ergodicity for Wigner matrices

Abstract

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